Quotient Rule of Exponents

The quotient rule of exponents states that when dividing exponential expressions with the same base, the base is retained, and the exponent in the denominator is subtracted from the exponent in the numerator.

This rule makes it easier to compare exponential growth without calculating large powers.

For example, a bacterial culture starts with x bacterial cells. The population after n hours is x times 2 to the power of n. So, it doubles each hour, becoming 2x, 4x, 8x, and so on.

Suppose the bacterial population is measured after 8 hours and then again after 12 hours.

The factor by which the bacterial population increases during those 4 hours can be found by dividing the count after 12 hours by the count after 8 hours. The initial number of cells, x, cancels out, leaving 2 to the power of 12 divided by 2 to the power of 8.

Using the quotient rule, the exponents are subtracted, leaving 2 to the power of 4, which equals 16. So, the population grew sixteen times during those four hours. This example shows how the quotient rule simplifies comparisons of exponential growth.